Mathematical Modelling and Applications

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Mathematical Modelling of the Co-Infection Dynamics of Typhoid Fever with Plasmodium Vivax and Plasmodium Falciparum with Treatment

Received: 26 December 2021    Accepted: 15 January 2022    Published: 24 January 2022
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Abstract

Malaria and typhoid fever are infectious and communicable diseases. Malaria remains one of the largest killer diseases in the world caused by one or more species of plasmodium parasites. Typhoid fever, also known as enteric fever, is a systemic bacterial infection disease caused by a highly virulent and invasive Salmonella enterica serovar Typhi (S. Typhi). Malaria and typhoid fever co-infection is the most endemic disease and a major public health problem in many tropical developing countries. Both diseases share similar transmission factor and often have the similar symptom. Because of the high prevalence of malaria and typhoid fever in these developing countries, co-infections are common. In addition to true co-infection of malaria and typhoid fever, the major challenges on managing controlling these diseases are false diagnoses due to similar signs and symptoms and false positive results in testing methods. In this study, we have formulated a mathematical model based on a system of non-linear first order ordinary differential equations to study the dynamics of the co-infection dynamics of plasmodium vivax- typhoid fever and plasmodium falciparum -typhoid fever. We have proved the existence of the disease free and endemic equilibrium points of the model and we used a non-linear stability analysis method to prove the local and global stabilities of these equilibrium points. Further, the positivity and boundedness of the solution of the model developed is verified to discover that the model equation is mathematically and epidemiologically well posed. We obtained the basic reproduction number R0 for the co-infection dynamics of plasmodium vivax, plasmodium falciparum and typhoid fever diseases in terms of the three basic reproduction numbers of the separate diseases using the standard data obtained from different sources. The separate diseases disappear from the community whenever the reproduction number R0 is very small and less than unity. On the other hand, the diseases co-exist whenever their reproduction numbers exceed unity (regardless which of the numbers is larger). The sensitivity analysis is discussed in detail to identify the most influential parameters that enhance the co-infection of malaria and typhoid fever disease in a given population. Numerical simulation is also done to illustrate the influence of different parameters on the basic reproduction number.

DOI 10.11648/j.mma.20220701.11
Published in Mathematical Modelling and Applications (Volume 7, Issue 1, March 2022)
Page(s) 1-25
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Infectious Disease, Typhoid Fever, Malaria, Plasmodium Falciparum, Plasmodium Vivax, Co-infection

References
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    Zeleke Amare Workie, Purnachandra Rao Koya. (2022). Mathematical Modelling of the Co-Infection Dynamics of Typhoid Fever with Plasmodium Vivax and Plasmodium Falciparum with Treatment. Mathematical Modelling and Applications, 7(1), 1-25. https://doi.org/10.11648/j.mma.20220701.11

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    ACS Style

    Zeleke Amare Workie; Purnachandra Rao Koya. Mathematical Modelling of the Co-Infection Dynamics of Typhoid Fever with Plasmodium Vivax and Plasmodium Falciparum with Treatment. Math. Model. Appl. 2022, 7(1), 1-25. doi: 10.11648/j.mma.20220701.11

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    AMA Style

    Zeleke Amare Workie, Purnachandra Rao Koya. Mathematical Modelling of the Co-Infection Dynamics of Typhoid Fever with Plasmodium Vivax and Plasmodium Falciparum with Treatment. Math Model Appl. 2022;7(1):1-25. doi: 10.11648/j.mma.20220701.11

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  • @article{10.11648/j.mma.20220701.11,
      author = {Zeleke Amare Workie and Purnachandra Rao Koya},
      title = {Mathematical Modelling of the Co-Infection Dynamics of Typhoid Fever with Plasmodium Vivax and Plasmodium Falciparum with Treatment},
      journal = {Mathematical Modelling and Applications},
      volume = {7},
      number = {1},
      pages = {1-25},
      doi = {10.11648/j.mma.20220701.11},
      url = {https://doi.org/10.11648/j.mma.20220701.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20220701.11},
      abstract = {Malaria and typhoid fever are infectious and communicable diseases. Malaria remains one of the largest killer diseases in the world caused by one or more species of plasmodium parasites. Typhoid fever, also known as enteric fever, is a systemic bacterial infection disease caused by a highly virulent and invasive Salmonella enterica serovar Typhi (S. Typhi). Malaria and typhoid fever co-infection is the most endemic disease and a major public health problem in many tropical developing countries. Both diseases share similar transmission factor and often have the similar symptom. Because of the high prevalence of malaria and typhoid fever in these developing countries, co-infections are common. In addition to true co-infection of malaria and typhoid fever, the major challenges on managing controlling these diseases are false diagnoses due to similar signs and symptoms and false positive results in testing methods. In this study, we have formulated a mathematical model based on a system of non-linear first order ordinary differential equations to study the dynamics of the co-infection dynamics of plasmodium vivax- typhoid fever and plasmodium falciparum -typhoid fever. We have proved the existence of the disease free and endemic equilibrium points of the model and we used a non-linear stability analysis method to prove the local and global stabilities of these equilibrium points. Further, the positivity and boundedness of the solution of the model developed is verified to discover that the model equation is mathematically and epidemiologically well posed. We obtained the basic reproduction number R0 for the co-infection dynamics of plasmodium vivax, plasmodium falciparum and typhoid fever diseases in terms of the three basic reproduction numbers of the separate diseases using the standard data obtained from different sources. The separate diseases disappear from the community whenever the reproduction number R0 is very small and less than unity. On the other hand, the diseases co-exist whenever their reproduction numbers exceed unity (regardless which of the numbers is larger). The sensitivity analysis is discussed in detail to identify the most influential parameters that enhance the co-infection of malaria and typhoid fever disease in a given population. Numerical simulation is also done to illustrate the influence of different parameters on the basic reproduction number.},
     year = {2022}
    }
    

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  • TY  - JOUR
    T1  - Mathematical Modelling of the Co-Infection Dynamics of Typhoid Fever with Plasmodium Vivax and Plasmodium Falciparum with Treatment
    AU  - Zeleke Amare Workie
    AU  - Purnachandra Rao Koya
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    T2  - Mathematical Modelling and Applications
    JF  - Mathematical Modelling and Applications
    JO  - Mathematical Modelling and Applications
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    EP  - 25
    PB  - Science Publishing Group
    SN  - 2575-1794
    UR  - https://doi.org/10.11648/j.mma.20220701.11
    AB  - Malaria and typhoid fever are infectious and communicable diseases. Malaria remains one of the largest killer diseases in the world caused by one or more species of plasmodium parasites. Typhoid fever, also known as enteric fever, is a systemic bacterial infection disease caused by a highly virulent and invasive Salmonella enterica serovar Typhi (S. Typhi). Malaria and typhoid fever co-infection is the most endemic disease and a major public health problem in many tropical developing countries. Both diseases share similar transmission factor and often have the similar symptom. Because of the high prevalence of malaria and typhoid fever in these developing countries, co-infections are common. In addition to true co-infection of malaria and typhoid fever, the major challenges on managing controlling these diseases are false diagnoses due to similar signs and symptoms and false positive results in testing methods. In this study, we have formulated a mathematical model based on a system of non-linear first order ordinary differential equations to study the dynamics of the co-infection dynamics of plasmodium vivax- typhoid fever and plasmodium falciparum -typhoid fever. We have proved the existence of the disease free and endemic equilibrium points of the model and we used a non-linear stability analysis method to prove the local and global stabilities of these equilibrium points. Further, the positivity and boundedness of the solution of the model developed is verified to discover that the model equation is mathematically and epidemiologically well posed. We obtained the basic reproduction number R0 for the co-infection dynamics of plasmodium vivax, plasmodium falciparum and typhoid fever diseases in terms of the three basic reproduction numbers of the separate diseases using the standard data obtained from different sources. The separate diseases disappear from the community whenever the reproduction number R0 is very small and less than unity. On the other hand, the diseases co-exist whenever their reproduction numbers exceed unity (regardless which of the numbers is larger). The sensitivity analysis is discussed in detail to identify the most influential parameters that enhance the co-infection of malaria and typhoid fever disease in a given population. Numerical simulation is also done to illustrate the influence of different parameters on the basic reproduction number.
    VL  - 7
    IS  - 1
    ER  - 

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Author Information
  • Department of Mathematics, Debre Berhan University, Debre Berhan, Ethiopia

  • Department of Mathematics, Wollega University, Nekemte, Ethiopia

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